# 二维积分 > 平面积分和曲线积分 ## 1 第一类曲线积分 $$ \begin{aligned} & {L=\mathop{{L}}\nolimits_{{1}}+\mathop{{L}}\nolimits_{{2}} \Rightarrow \mathop{ \int }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s=\mathop{ \int }\nolimits_{{\mathop{{L}}\nolimits_{{1}}}}f{ \left( {x,y} \right) } \text{d} s+\mathop{ \int }\nolimits_{{\mathop{{L}}\nolimits_{{2}}}}f{ \left( {x,y} \right) } \text{d} s}\\ & {\mathop{ \iint }\nolimits_{{L}}{ \left[ { \alpha f{ \left( {x,y} \right) }+ \beta f{ \left( {x,y} \right) }} \right] } \text{d} s= \alpha \mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s+ \beta \mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s}\\ & {f{ \left( {x,y} \right) } \le g{ \left( {x,y} \right) } \Rightarrow \mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s \le \mathop{ \iint }\nolimits_{{L}}g{ \left( {x,y} \right) } \text{d} s}\\ &{ \left| {\mathop{ \iint }\nolimits_{{L}}f{ \left( {x,y} \right) } \text{d} s} \left| \le \mathop{ \iint }\nolimits_{{L}}{ \left| {f{ \left( {x,y} \right) }} \right| } \text{d} s\right. \right. } \end{aligned} $$ ## 2 对弧长的曲线积分 $$ \begin{aligned} \end{aligned} $$ ## 3 二重积分
